Deck 10: Additional Topics in Probability

Question 1

An urn contains 20 marbles,each of which shows a number.Eight marbles show 1,five marbles show 2,and seven marbles show 3.A marble is randomly selected and the number X that shows is observed.The mean of X is

A) 1.
B)
C)
D)
E)

Answer 1

Question 2

From a group of 10 men and 10 women,two people are randomly selected to form a committee.If X is the number of women on the committee,then P(X = 2)=

A)
B)
C)
D)
E)

Answer 2

Question 3

An urn contains three red and one green marble.Two marbles are randomly drawn in succession without replacement.If X is the number of red marbles that are drawn and f is the distribution of X,find f(0),f(1),and f(2).

Answer 3

f(0)= 0; f(1)= ; f(2)=

Question 4

A random variable X has a distribution given by f(0)= 0.3,f(1)= 0.2,f(2)= 0.5.(a)Find μ.(b)Find Var(X).

Question 5

A random variable X has a distribution given by f(1)= 0.6,f(2)= 0.2,f(3)= 0.2.The mean of X is

A) 0.33.
B) 0.4.
C) 1.6.
D) 1.8.
E) 3.

Question 6

Suppose that you pay $3 to play a game in which a fair die is rolled.You receive the number of dollars equal to the number of dots that appear.What is your expected gain (or loss)on each play?

Question 7

An insurance company offers a life policy to individuals in a certain group.For a given 1-year period,the company will pay $5000 in the event of the death of a policyholder.If the probability that a person in this group dies during a 1-year time period is 0.001 and the annual premium for the policy is $10,then the expected gain per policy to the company is

A) $4.99.
B) $5.00.
C) $9.90.
D) $4990.
E) $4994.99.

Question 8

n = 42; p = 0.2

A) σ = 2.59
B) σ = 5.86
C) σ = 0.18
D) σ = 6.71

Question 9

n = 46; p =

A) σ = 3.32
B) σ = 6.59
C) σ = 0.91
D) σ = 7.44

Question 10

A random variance X has a distribution given by f(0)= 0.4,f(1)= 0.3,f(2)= 0.3.The variance of X is

A) 0.69.
B) 0.54.
C) 0.50.
D) 0.47.
E) 0.32.

Question 11

n = 46; p =

A) μ = 27.6
B) μ = 27.9
C) μ = 28.3
D) μ = 27.1

Question 12

P(X = 2); n = 5,p = 0.70

A) 0.132
B) 0.198
C) 0.700
D) 0.464

Question 13

An urn contains 4 red and 6 green marbles.Two marbles are successively drawn at random with replacement and the number of red marbles,X,is observed.Construct the probability histogram for X.

Question 14

From a standard deck of 52 playing cards,two cards are randomly drawn in succession without replacement.If X is the number of red cards that are drawn and f is the distribution of X,find f(0),f(1),and f(2).

Question 15

P(X = 2); n = 10,p =

A) 0.1951
B) 0.2156
C) 0.1929
D) 0.0028

Question 16

At the time of purchase of its product,a company offers a contract to its customers that guarantees the replacement of the product if it malfunctions within a one-year period.The cost of the contract is $10.The probability that a unit of the product malfunctions within one year of purchase is 0.06.If the cost to the company for replacement of a unit is $160,determine the company's expected gain or loss per contract.

Question 17

n = 20; p = 0.2

A) μ = 4.0
B) μ = 4.3
C) μ = 4.7
D) μ = 3.5

Question 18

Suppose that you pay $1 to play a game in which a pair of fair dice are rolled.For each 2 that shows on a die you receive $2.To the nearest cent,your expected loss on each play is

A) $0.67.
B) $0.33.
C) $0.25.
D) $0.15.
E) $0.07.

Question 19

P(X = 3); n = 4,p =

A) 0.0154
B) 0.0116
C) 0.0039
D) 0.0231

Question 20

A random variable X has a distribution given by f(0)= 0.3,f(1)= 0.6,f(2)= 0.1.(a)Find μ.(b)Find σ.

Question 21

If a fair coin is tossed six times,determine the probability that at least two heads show.

Question 22

A manufacturer produces thermostats,10% of which are defective.From a production run,a sample of four thermostats is randomly selected.Determine the probability that at least three of the selected thermostats are defective.You may assume that the four trials are independent and that the number of defective items in the sample has a binomial distribution.

A) 0.0035
B) 0.0036
C) 0.0037
D) 0.0038
E) 0.0039

Question 23

Let X be the number of smokers who quit cold turkey.The probability of quitting cold turkey is 10%.For a group of 5 smokers,selected independently,find the distribution of X.

Question 24

For the transition matrix T =
,find the probability of going from state 2 to state 1 after three steps.

Question 25

Find the steady state vector for the transition matrix T =
.

Question 26

P(X = 6); n = 13,p = 0.5

A) 0.2095
B) 0.3142
C) 0.2723
D) 0.0156

Question 27

If
is a transition matrix for a Markov chain,determine the value of b.

Question 28

If is a transition matrix for a Markov chain,determine the values of a,b,and c.

Question 29

A platypus net is set out on four successive days where a capture-recapture study is located.Only 1 platypus can be netted per day,and the probability that a platypus will enter the trap and be netted is
.If X is the number of platypuses captured over the four days,find the distribution of X.

Question 30

If a person watches a certain TV daily evening news program on one evening,then the probability that the person watches that program the next evening is 0.7.However,if the person does not watch the program one evening,then the probability that the person watches the program the next evening is 0.2.
(a)If the person watches the program on Monday,what is the probability that the person watches the program on Wednesday?
(b)If 20% of the population watches the program on Thursday,what percentage can be expected to watch on Friday?

Question 31

A fair die is rolled four times.What is the probability that exactly three 2's appear?

Question 32

For the transition matrix T =
,find the probability of going from state 1 to state 2 after two steps.

Question 33

If
=
is an initial state vector for the transition matrix T =
,compute the state vector
.

Question 34

An urn contains two red and eight green marbles.Four marbles are randomly drawn in succession with replacement.Determine the probability that exactly two marbles are red.

Question 35

A manufacturer produces light bulbs,of which 3% are defective.From a production run of 5000 bulbs,three bulbs are randomly selected and each is tested.Find,to three decimal places,the probability that the sample contains exactly one defective bulb.Assume that the three trials are independent and that the number of defective bulbs in the sample has a binomial distribution.

Question 36

25% of a video store's customers are female.Let X be the number of women among the next 5 customers,and find the distribution of X.

Question 37

Find the steady state vector for the transition matrix T =
.

Question 38

If X is a binomial random variable involved with six independent trials,and the probability of success on any trial is ,then the probability that X = 2 is

A)
B)
C)
D)
E)

Question 39

A biased coin is tossed 8 times.If the probability of heads appearing on any toss is ,then the probability that exactly six heads appear is

A)
B)
C)
D)
E)

Question 40

If
=
is an initial state vector for the transition matrix T =
,compute the state vector
.

Question 41

If is a transitional matrix for a Markov chain,determine the value of b.

A) 0.1
B) 0.2
C) 0.4
D) 0.6
E) 0.8

Question 42

For the matrix T = ,find the probability of going from state 2 to state 1 after two steps.

A)
B)
C)
D)
E)

Question 43

A college dining hall has available two fruit juices for breakfast: orange and grapefruit.One hundred students drink juice on a daily basis.It is found that a student will not drink grapefruit juice on two successive days,and if a student drinks orange juice one day,then the following day the student is equally likely to drink orange juice as to drink grapefruit juice.If 40 of the regular juice drinkers drink orange juice on Monday,how many can be expected to drink orange juice on Wednesday?

A) 10
B) 30
C) 50
D) 60
E) 90

Question 44

If =
Is an initial state vector for the transition matrix T =
,compute the state vector
)

A)
B)
C)
D)
E)

Question 45

Suppose T = is the transition matrix for a Markov chain.If Q =
Is the steady-state vector,find

A)
B)
C)
D)
E)